Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 1, pp. 157-178.

This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-Stokes system generalizing the prototype

{n t +u·n=Δn m -·(nc),c t +u·c=Δc-nc,u t +P=Δu+nφ,·u=0,
which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid.It is proved that global weak solutions exist whenever m>8 7 and the initial data (n 0 ,c 0 ,u 0 ) are sufficiently regular satisfying n 0 >0 and c 0 >0. This extends a recent result by Di Francesco, Lorz and Markowich [M. Di Francesco, A. Lorz, P.A. Markowich, Chemotaxis–fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst. Ser. A 28 (2010) 1437–1453] which asserts global existence of weak solutions under the constraint m[7+217 12,2].

Ce papier considère un problème aux limites dans des domaines tridimensionnels réguliers et bornés, plus précisément, un système couplé de chemotaxie-Stokes qui généralise le prototype

{n t +u·n=Δn m -·(nc),c t +u·c=Δc-nc,u t +P=Δu+nφ,·u=0
et qui décrit le mouvement des bactéries nageuses conduites par lʼoxygène dans un fluide incompressible.On montre que les solutions faibles globales existent quand m>8 7 et la donnée initiale (n 0 ,c 0 ,u 0 ) est suffisamment régulière et vérifie n 0 >0 et c 0 >0. Cela étend le résultat récent de Di Francesco, Lorz et Markowich [M. Di Francesco, A. Lorz, P.A. Markowich, Chemotaxis–fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst. Ser. A 28 (2010) 1437–1453] qui affirme lʼexistence globale de solutions faibles sous la contrainte m[7+217 12,2].

DOI: 10.1016/j.anihpc.2012.07.002
Classification: 35K55, 35Q92, 35Q35, 92C17
Keywords: Chemotaxis, Stokes, Nonlinear diffusion, Global existence, Boundedness
Keywords: Chemotaxie, Stokes, Diffusion nonlinéaire, Existence globale, Estimation uniforme
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     title = {Locally bounded global solutions in a three-dimensional {chemotaxis-Stokes} system with nonlinear diffusion},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     publisher = {Elsevier},
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Tao, Youshan; Winkler, Michael. Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 1, pp. 157-178. doi : 10.1016/j.anihpc.2012.07.002. http://www.numdam.org/articles/10.1016/j.anihpc.2012.07.002/

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